$f_{X_{(k)}}(x) = \frac{n!}{(k-1)!(n-k)!}\,[F(x)]^{k-1}[1-F(x)]^{n-k}f(x)$ $\text{where } X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)} \text{ are the sorted values of the sample}$ The $k$-th Order Statistic $X_{(k)}$ is the $k$-th smallest value in a random sample of $n$ independent and identically distributed random variables with common PDF $f$ and CDF $F$. The formula counts the ways exactly $k-1$ observations fall below $x$, one falls at $x$, and $n-k$ fall above $x$. > [!example]- Minimum Lifetime of Three Machines {💡 Example} > Three machines have independent Exp(1) lifetimes. Find the PDF of the minimum lifetime $X_{(1)}$. > > > [!answer]- Answer > > With $n=3$, $k=1$, $f(x)=e^{-x}$, $F(x)=1-e^{-x}$: > > $f_{X_{(1)}}(x) = \frac{3!}{0!\,2!}\,[1-e^{-x}]^0\,[e^{-x}]^2\,e^{-x} = 3e^{-3x}, \quad x>0$ > > This is an Exp(3) distribution, confirming that the minimum of $n$ independent Exp($\lambda$) variables is Exp($n\lambda$).